Vol 7, No 8 (2016)

(1)

Available online through

ISSN 2229 – 5046

VAGUE GENERALIZED ALPHA CLOSED SETS IN TOPOLOGICAL SPACES

L. MARIAPRESENTI*

1

_{, I. AROCKIARANI}

2

1,2_{Nirmala college for women, Coimbatore-641018, Tamilnadu, India.}

(Received On: 28-07-16; Revised & Accepted On: 12-08-16)

ABSTRACT

T

he aim of this paper is to explore the notion of vague sets to define a new class of generalized sets namely vague

generalized α-closed sets and investigate their properties.

Keywords: Vague set, Vague topology, Vague generalized alpha closed set.

1. INTRODUCTION

Classical mathematical methods are not enough to solve the problems of daily life and also are not enough to meet the new requirements. Therefore, some theories such as Fuzzy set theory [19], Rough set theory [17], Soft set theory [15] and Vague set theory [5]. have been developed to solve these problems.

Applications of these theories appear in topology and in many areas of mathematics. Most of these problems were solved by fuzzy set provided by Zadeh [19] and later Atanassov [2] generalized this idea to the new class of intuitionistic fuzzy sets using the notion of fuzzy sets. The theory of vague sets was first proposed by Gau and Buehrer [7]. Vague set theory is actually an extension of fuzzy set theory and vague sets are regarded as a special case of context- dependent fuzzy sets.

2. PRELIMINARIES

Definition 2.1 [4]: A vague set A in the universe of discourse U is characterized by two membership functions given by:

(i) A true membership function

t

_A

:

U

→

[

0 ,

1 ]

and (ii) A false membership function

f

A

:

U

→

[

0 ,

1 ]

where

t

_A

(

x

)

is a lower bound on the grade of membership of x derived from the “evidence for x”,

f

_A

(

x

)

is a lower bound on the negation of x derived from the “evidence for x”, and

t

_A

(

x

)

+

f

_A

(

x

)

≤

1

. Thus the grade of membership of u in the vague set A is bounded by a subinterval

[

t

A

(

x

),

1 −

f

A

(

x

)]

of [0,1]. this indicates that if the actual grade of

membership of x is µ(x), then,

t

_A

(

x

)

≤

µ

(

x

)

≤

1 −

f

_A

(

x

)

.The vague set A is written as

[

]

{

x

t

x

f

x

u

U

}

A

=

,

A

(

),

1 −

A

(

)

/

∈

where the interval

[

t

A

(

x

),

1 −

f

A

(

x

)]

is called the vague value of x in A,

denoted by

V

A

(

x

)

.

Definition 2.2 [7]: Let A and B be VSs of the form

A

=

{

x

,

[

t

A

(

x

),

1 −

f

A

(

x

)

]

/

x

∈

X

}

and

[

]

{

x

t

x

f

x

X

}

B

=

,

B

(

),

1 −

B

(

)

/

∈

Then

(i)

A

⊆

B

if and only if

t

_A

(

x

)

≤

t

_B

(

x

)

and

1 −

f

_A

(

x

)

≤

1 −

f

_B

(

x

)

for all x

∈

X

(ii)

A=B if and only if

A

⊆

B

and

B

⊆

A

(iii)

A

{

x

f

A

x

t

A

x

X

}

c

=

−

∈

/

)

(

1 ),

(

,

(iv)

A



B

=

{

x

,

min

(

t

A

(

x

),

t

B

(

x

)

,

min

(

1 −

f

A

(

x

),

1 −

f

B

(

x

)

/

x

∈

X

}

(v)

AB=

{

x,max

(

t_A(x),t_B(x)

)

,max

(

1− f_A(x),1− f_B(x)

)

/x∈X

}

(2)

For the sake of simplicity, we shall use the notation

A

=

x

,

t

_A

,

1 −

f

_A instead of

[

]

{

x

t

x

f

x

X

}

A

=

,

A

(

),

1 −

A

(

)

/

∈

.

Definition 2.3: A subset A of a topological space (X,

τ

) is called (i) a preclosed set [13] if

cl

(int(

A

))

⊆

A

(ii) a semi-closed set[9] if

int(

cl

(

A

))

⊆

A

(iii)a regular closed set[18] if

A

=

cl

(

int

( )

A

)

(iv)a α-closed set [14] if

cl

(int(

cl

(

A

)))

⊆

A

Definition 2.4: A subset A of a topological space (X,

τ

) is called

(i) a generalized closed set (briefly g-closed)[8] if

cl

(

A

)

⊆

U

whenever

A

⊆

U

and U is an open set in X. (ii) a semi-generalized closed set (briefly sg-closed) [3] if

scl

(

A

)

⊆

U

whenever

A

⊆

U

and U is semi-open set

in X

(iii)a generalized semi-closed set (briefly gs-closed) [1]if

scl

(

A

)

⊆

U

whenever

A

⊆

U

and U is open set in X

(iv)a generalized semi pre closed set (briefly gsp-closed) [6]if

spcl

(

A

)

⊆

U

whenever

A

⊆

U

and U is open

set in X

(v) a generalized pre closed set (briefly gp-closed) [4]if

pcl

(

A

)

⊆

U

whenever

A

⊆

U

and U is open set in X

(vi) a generalized α-closed set (briefly gα-closed) [11] if

α

cl

(

A

)

⊆

U

whenever

A

⊆

U

and U is α-open set in X.

(vii) a α-generalized closed set (briefly αg-closed) [10]

α

cl

(

A

)

⊆

U

whenever

A

⊆

U

and U is open set in X. 3. VAGUE TOPOLOGICAL SPACE

Definition 3.1: A vague topology (VT in short) on X is a family

τ

of VSs in X satisfying the following axioms. (i) 0, 1

∈

τ

(ii)

G

₁



G

₂

∈

τ

,

for any

G

₁

,

G

₂

∈

τ

(iii)



G

_i

∈

τ

for any family

{

G

_i

/

i

∈

J

}

⊆

τ

.

In this case the pair

(

X

,

τ

)

is called a Vague topological space (VTS in short) and any VS in

τ

is known as a Vague open set (VOS in short) in X.

The complement Ac of a VOS A in a VTS (X,

τ

) is called a vague closed set (VCS in short) in X.

Definition 3.2: Let (X,

τ

) be a VTS and

A

=

x

,

t

_A

,

1 −

f

_A be a VS in X. Then the vague interior and a vague closure are defined by

Vint(A)=



{G/G is an VOS in X and G

⊆

A}

Vcl(A)=



{K/K is an VCS in X and A

⊆

K}

Note that for any VS A in

(

X

,

τ

)

, we have

Vcl

(

A

c

)

=

(

V

int(

A

))

cand

V

int(

A

c

)

=

(

Vcl

(

A

))

c

.

Example 3.3: We consider the VT Let Let X={a,b} and let

τ

=

{

0 ,

G

,

1 }

is an VT on X, where

[

] [

]

{

x

,

0 .

1 ,

0 .

5 ,

0 .

1 ,

0 .

6 }

G

=

. Here the only open sets are 0, 1, and G. If

A

=

{

x

,

[

0 .

1 ,

0 .

6 ] [

,

0 .

1 ,

0 .

9 ]

}

is a VT

on X then,

Vint(A)=



⊆

A}=G

Vcl(A)=



⊆

K}=Gc

Definition 3.4: A vague set A of

(

X

,

τ

)

is said to be a, (i) a vague pre-closed set if

Vcl

(

V

int(

A

))

⊆

A

(ii) a vague semi-closed set if

V

int(

Vcl

(

A

))

⊆

A

(iii)a vague regular-closed set if

A

=

Vcl

(

V

int

( )

A

)

(3)

Definition 3.5: An vague set A in

(

X

,

τ

)

is said to be a,

(i) vague generalized closed set (briefly VGC) if

Vcl

(

A

)

⊆

U

whenever

A

⊆

U

and U is a vague open set in X.

(ii) vague generalized semi-closed set (briefly VGSC) if

Vscl

(

A

)

⊆

U

_whenever

A

⊆

U

and U is vague open

set in X.

(iii)vague generalized pre closed set (briefly VGPC) if

Vpcl

(

A

)

⊆

U

whenever

A

⊆

U

and U is a vague open

set in X.

Properties 3.6: Let A be any Vague set in

(

X

,

τ

)

. then (i)

V

int(

1 −

A

)

=

1 −

(

Vcl

( )

A

)

and

(ii)

Vcl

(

1 −

A

)

=

1 −

(

V

int

( )

A

)

Proof: (i) By definition Vcl(A)=



⊆

K}.

( )

(

Vcl

A

)

−

1

=1-



⊆

K}.

=



{1 - K/K is an VCS in X and A

⊆

K}. =



⊆

1- A} =

V

int(

1 −

A

)

(ii) The proof is similar to (i).

Theorem 3.7: Let

(

X

,

τ

)

be a VS and let

A

∈

V

(

X

).

Then the following properties hold. (i)

V

int(

A

)

⊂

A

(ii)

A

⊂

B

⇒

V

int(

A

)

⊂

V

int(

B

)

(iii)

V

int(

A

)

∈

τ

(iv)

A

is a vague open set

⇔

V

int(

A

)

=

A

(v)

V

int(

V

int(

A

))

=

V

int(

A

)

(vi)

V

int(

0 )

=

0 ,

V

int(

1 )

=

1

(vii)

V

int(

A



B

)

=

V

int(

A

)



V

int(

B

)

(viii)

(

V

int(

A

))

c

=

Vcl

(

A

c

)

Proof: The proof is obvious.

(

X

,

τ

)

be a VS and Let

A

∈

V

(

X

)

. Then the following properties hold.

(i)

(

A

)

⊂

Vcl

(

A

)

(ii)

A

⊂

B

⇒

Vcl

(

A

)

⊂

Vcl

(

B

)

(iii)

(

Vcl

(

A

))

C

∈

τ

(iv)

A

is a vague Closed set

⇔

Vcl

(

A

)

=

A

(v)

Vcl

(

Vcl

(

A

))

=

Vcl

(

A

)

(vi)

Vcl

(

0 )

=

0

,

Vcl

(

1 )

=

1

(vii)

Vcl

(

A



B

)

=

Vcl

(

A

)



Vcl

(

B

)

(viii)

(

Vcl

(

A

))

c

=

V

int(

A

c

)

Proof: The proof is obvious.

4. VAGUE GENERALIZED ALPHA CLOSED SETS

Definition 4.3: A vague set A in

(

X

,

τ

)

is said to be a vague generalized alpha closed set (VGαCS in short) if

U

A

cl

V

α

(

)

⊆

whenever

A

⊆

U

and U is a VαOS in

(

X

,

τ

)

Definition 4.4: A vague set A in

(

X

,

τ

)

is said to be a vague alpha generalized closed set (VGαCS in short) if

U

A

cl

(4)

Example 4.5: Let X={a, b} and let

τ

=

{

0 ,

G

,

1 }

G

=

{

x

,

[

0 .

4 ,

0 .

8 ] [

,

0 .

3 ,

0 .

7 ]

}

. Here the only

α open sets are 0, X, and G. Then the VS

A

=

{

x

,

[

0 .

4 ,

0 .

9 ] [

,

0 .

4 ,

0 .

8 ]

}

is an VGαCS in

(

X

,

τ

)

.

Theorem 4.6: For any VTS

(

X

,

τ

)

, we have the following:

(i)

Every VCS is a VGαCS in X

(ii)

Every VαCS is a VGαCS in X

(iii)

Every VRCS is a VGαCS in X

(iv)

Every VGαCS is a VαGCS in X

(v)

Every VGαCS is a VGPCS in X

(vi)

Every VGαCS is a VGSCS in X.

Proof:

(i) Let A be a VCS. Let

A

⊆

U

and U is a VαOS in

(

X

,

τ

)

. Since

V

α

cl

(

A

)

⊆

Vcl

(

A

)

and A is a VCS,

U

A

Vcl

A

cl

V

α

(

)

⊆

(

)

=

⊆

. Therefore A is a VGαCS in X.

(ii) Let

A

⊆

U

and U is a VαOS in

(

X

,

τ

)

.By hypothesis

V

α

cl

(

A

)

=

A

. Hence

V

α

cl

(

A

)

⊆

U

. Therefore A

is a VGαCS in X

(iii)(iii), (iv), (v) and (vi)are obvious.

But converse of the above need not be true. It can be shown by the following example.

Example 4.7: Let X ={a, b} and let

τ

=

{

0 ,

G

,

1 }

is a VT on X where

G

=

{

x

, 0.2, 0.7 , 0.4, 0.6

[

] [

]

}

.

Let

[

] [

]

{

x

,

0 .

3 ,

0 .

8 ,

0 .

5 ,

0 .

7 }

A

=

be any VS in X.Here

V

α

cl

(

A

)

⊆

G

, whenever

A

⊆

G

, for all VαOS G in X. A is

a VGαCS, but not a VCS in X, since

Vcl

(

A

)

=

1 ≠

A

.

Example 4.8: Let X ={a,b} and let

τ

=

{

0 ,

G

,

1 }

is an VT on X where

G

=

{

x

, 0.2, 0.6 , 0.3, 0.5

[

] [

]

}

.

Let

[

] [

]

{

x

,

0 .

4 ,

0 .

9 ,

0 .

4 ,

0 .

8 }

A

=

be any VS in X. Clearly

V

α

cl

(

A

)

⊆

G

, whenever

A

⊆

G

, for all VαOS G in X.

Therefore A is an VGαCS, but not a VαCS in X, since

Vcl

(

V

int

(

Vcl

( )

A

)

=

1 ⊄

A

.

Example: 4.9: Let X ={a,b} and let

τ

=

{

0 ,

G

,

1 }

is a VT on X where

G

=

{

x

, 0.1, 0.5 , 0.3, 0.7

[

] [

]

}

.

Let

[

] [

]

{

x

,

0 .

2 ,

0 .

6 ,

0 .

4 ,

0 .

8 }

A

=

be any VS in X. A is a VGαCS, but not a VRCS in

(

X

,

τ

)

, since

( )

(

V

int

A

)

G

A

.

Vcl

=

c

≠

τ

=

{

0 ,

G

,

1 }

is a VT on X where

G

=

{

x

,

[

0 .

2 ,

0 .

7 ] [

,

0 .

3 ,

0 .

5 ]

}

and let

[

] [

]

{

x

,

0 .

1 ,

0 .

6 ,

0 .

2 ,

0 .

5 }

A

=

be any VS in X. Here A is a VαGCS in X. Consider the VαOS

[

] [

]

{

,

0 .

3 ,

0 .

8 ,

0 .

4 ,

0 .

7 }

1

x

G

=

. Here A is a VαGCS in X.

Example 4.11: Let X ={a, b} and let

τ

=

{

0 ,

G

,

1 }

is a VT on X where

G

=

{

x

,

[

0 .

3 ,

0 .

6 ] [

,

0 .

2 ,

0 .

8 ]

}

Here the only

α open sets are 0,X and G. Let

A

=

{

x

,

[

0 .

3 ,

0 .

6 ] [

,

0 .

3 ,

0 .

5 ]

}

. be any VS in X. Here

Vpcl

(

A

)

=

1 ⊆

1

. Therefore A

is a VGPCS in

(

X

,

τ

)

, but not a VGαCS.

(5)

The diagram gives the implication of the above theorems:

Remark 4.13: VPCS and VGαCS are independent to each other.

τ

=

{

0 ,

G

,

1 }

is a VT on X, where

G

=

{

x

,

[

0 .

3 ,

0 .

6 ] [

,

0 .

1 ,

0 .

7 ]

}

. Here the only

α open sets are 0,1 and G. Let

A

=

{

x

,

[

0 .

4 ,

0 .

7 ] [

,

0 .

3 ,

0 .

8 ]

}

. be a VGαCS(X) but not a VPCS(X), since

A

G

A

V

Vcl

(

int(

))

=

c

⊄

.

τ

=

{

0 ,

G

,

1 }

is a VT on X, where

G

=

{

x

,

[

0 .

1 ,

0 .

7 ] [

,

0 .

3 ,

0 .

8 ]

}

. Here the only

α open sets are 0, 1 and G. Let

A

=

{

x

,

[

0 .

1 ,

0 .

6 ] [

,

0 .

2 ,

0 .

7 ]

}

. be a VPCS(X) but not a VGαCS (X).

Remark 4.16: VSCS and VGαCS are independent to each other.

τ

=

{

0 ,

G

,

X

}

is a VT on X, where

G

=

{

x

,

[

0 .

4 ,

0 .

6 ] [

,

0 .

3 ,

0 .

6 ]

}

. Let

[

] [

]

{

x

,

0 .

5 ,

0 .

7 ,

0 .

5 ,

0 .

8 }

A

=

be an VGαCS (X) but not a VSCS (X), since

V

int(

Vcl

(

A

))

=

1 ⊄

A

.

τ

=

{

0 ,

G

,

1 }

is a VT on X, where

G

=

{

x

,

[

0 .

1 ,

0 .

6 ] [

,

0 .

2 ,

0 .

7 ]

}

. Let

[

] [

]

{

x

,

0 .

1 ,

0 .

7 ,

0 .

2 ,

0 .

7 }

A

=

be a VSCS (X) but not a VGαCS (X), since

V

α

cl

(

A

)

=

G

c

⊄

G

.

(

X

,

τ

)

be a VTS. Then for every

A

∈

VG

α

C

(

X

)

and for every

B

∈

VS

(

X

)

,

)

(

A

cl

V

B

A

⊆

α

implies

B

∈

VG

α

C

(

X

)

.

Proof: Let a VS

B

⊆

U

and U be a Vαos in X. Since

A

⊆

B

,

A

⊆

U

and A is VGαCS,

V

α

cl

(

A

)

⊆

U

. By hypothesis,

B

⊆

V

α

cl

(

A

),

V

α

cl

(

B

)

⊆

V

α

cl

(

A

)

⊆

U

.

Therefore

V

α

cl

(

B

)

⊆

U

. Hence B is VGαCS of X. Remark 4.20: The intersection of any two VGαCS is not VGαCS in general as seen from the following example.

τ

=

{

0 ,

G

,

1 }

is a VT on X, where

G

=

{

x

,

[

0 .

1 ,

0 .

7 ] [

,

0 .

3 ,

0 .

8 ]

}

. Then the VSs

[

] [

]

{

x

,

0 .

1 ,

0 .

8 ,

0 .

2 ,

0 .

8 }

(6)

Theorem 4.22: The union of two VGαCS is an VGαCS in

(

X

,

τ

)

, if they are VCS in

(

X

,

τ

)

Proof: Assume that A

and B are VGαCS in

(

X

,

τ

)

. Since A and B are VCS in X.

Vcl

(

A

)

=

A

and

Vcl

(

B

)

=

B

. Let

A



B

⊆

U

and

U is VαOS in X. Then

Vcl

(

V

int

(

Vcl

(

A



B

)

=

Vcl

(

V

int

(

A



B

)

⊆

Vcl

(

A



B

)

=

A



B

⊆

U

, that is

U

B

A

cl

V

α

(



)

⊆

Therefore

A



B

is VGαCS.

5. VAGUE GENERALIZED ALPHA OPEN SETS

Definition 5.1: A VS A is said to be a vague generalized alpha open set (VGαOS in short) in

(

X

,

τ

)

if the

complement

A

cis a VGαOS in X.

The family of all VGαOSs of a VTS

(

X

,

τ

)

is denoted by VGαO(X).

Theorem 5.2: For any VTS

(

X

,

τ

)

, we have the following:

(ii) Every VOS is a VGαOS

(iii)Every VαOS is a VGαOS

(iv)Every VROS is a VGαOS. But the converses need not be true.

Proof: (i) Let A be a VOS in X. Then

A

c is an VCS in X. Therefore by theorem 3.3

A

cis a VGαCS in X. Hence A is

a VGαOS in X.

The proof of (ii) and (iii) are obvious.

The converse of the above theorem is shown by the following example.

Example 5.3: Let X ={a, b} and let

τ

=

{

0 ,

G

,

1 }

is a VT on X, where

G

=

{

x

, 0.3, 0.6 , 0.2, 0.7

[

] [

]

}

.

Here the

only α open sets are 0, X and G. Let

A

=

{

x

,

[

0 .

3 ,

0 .

7 ] [

,

0 .

4 ,

0 .

8 ]

}

be any VS in X. A is a VGαOS, but not a VOS in

X.

τ

=

{

0 ,

G

,

1 }

is a VT on X, where

G

=

{

x

,

[

0 .

3 ,

0 .

5 ] [

,

0 .

2 ,

0 .

6 ]

}

Here the only α open sets are 0, X and G. Let

A

=

{

x

,

[

0 .

3 ,

0 .

6 ] [

,

0 .

3 ,

0 .

8 ]

}

be any VS in X. A is a VGαOS, but not a VαOS in X.

τ

=

{

0 ,

G

,

1 }

G

=

{

x

,

[

0 .

3 ,

0 .

6 ] [

,

0 .

2 ,

0 .

6 ]

}

Here the only α open sets are 0, X and G. Let

A

=

{

x

,

[

0 .

4 ,

0 .

7 ] [

,

0 .

3 ,

0 .

6 ]

}

be any VS in X. A is a VGαOS, but not a VROS in X.

(

X

,

τ

)

be an VTS. If A is an VS of X then the following properties are equivalent:

(i)

A

∈

VG

α

O

(

X

)

(ii)

V

⊆

v

int

(

Vcl

(

V

int

( )

A

)

whenever

V

⊆

A

and V is a VαCS in X.

(iii)

Thereexist VOS G1 such that

G

1

⊆

V

⊆

V

int

(

Vcl

( )

G

)

where G = Vint (A),

V

⊆

A

Proof: (i)

⇒

(ii) Let

A

∈

VG

α

O

(

X

)

. Then

A

cis a VGαCS in X. Therefore

V

α

cl

(

A

c

)

⊆

U

Whenever

A

c

⊆

U

and U is an VαOS in X. That is

(

( )

c

)

c

(

( )

c

)

c

A

Vcl

V

A

Vcl

V

Vcl

int

=

int

=

V

int

(

Vcl

(

Vcl

( )

A

c

)

c

)

( )

(

)

(

_c c

)

A

V

Vcl

V

int

=

(

( )

)

c

U

A

Vcl

V

⊇

=

int

. This implies

U

c

⊆

V

int

(

Vcl

(

Vcl

( )

A

)

whenever

A

U

c

⊆

and

U

cis a VαCS in X. Replace c

U

by V,

V

⊆

V

int

(

Vcl

(

V

int

( )

A

)

whenever

V

⊆

A

(ii)

⇒

(iii) Let

V

⊆

V

int

(

Vcl

(

V

int

( )

A

)

whenever

V

⊆

A

and V is a VαCS in X. Hence

( )

(

)

(

Vcl

V

A

)

V

int(

)

⊆

int

. Then there exist VOS G1 in X such that

G

1

⊆

V

⊆

V

int

(

Vcl

( )

G

)

where

(7)

(iii)

⇒

(i) Suppose that there exists VOS G1 such that

G

1

⊆

V

⊆

V

int

(

Vcl

( )

G

)

where G= Vint(A);

V

⊆

A

and V

is a VαOs in X. It is clear that

(

( )

)

c c

V

G

Vcl

V

int

⊆

. That is

(

V

int

(

Vcl

(

V

int

( )

A

)

c

⊆

V

c. This implies

( )

(

)

(

)

c c

V

A

V

Vcl

int

⊆

. Therefore

Vcl

(

V

int

(

Vcl

( )

A

c

)

⊆

V

c,

A

c

⊆

V

c and

V

cis VαOS in X. Hence

c c

V

A

cl

V

α

(

)

⊆

. That is

A

cis a VGαCS in X. This implies

A

∈

VG

α

O

(

X

)

.

Theorem 5.7: Let

(

X

,

τ

)

be a VTS. Then for every

A

∈

VG

α

O

(

X

)

and for every

B

∈

VS

(

X

)

,

A

B

A

V

α

int(

)

⊆

implies

B

∈

VG

α

O

(

X

)

.

Proof: By hypothesis

V

α

int(

A

)

⊆

B

⊆

A

. Taking complement on both sides, we get

A

c

⊆

B

c

⊆

(

V

α

int(

A

))

c.

Let

B

c

⊆

U

and U is VαOS in X. Since

A

c

⊆

B

c

,

A

c

⊆

U

and since

A

c is a VGαCS,

V

α

cl

(

A

c

)

⊆

U

, Also

))

(

))

int(

(

c c

c

A

cl

V

A

V

B

⊆

α

=

α

. Therefore

V

α

cl

(

B

c

)

⊆

V

α

cl

(

A

c

)

⊆

U

. Hence

B

cis a VGαCS in X. This

implies B is an VGαOS in X. That is

B

∈

VG

α

O

(

X

)

.

Theorem 5.8: A VS A of a VTS

(

X

,

τ

)

is a VGαOS if and only if

G

⊆

V

α

cl

( )

A

, whenever G is a VαCS(X) and

.

A

G

⊆

Proof: Necessity: Assume that A is a VGαOS in X. Also let G be a VαCS in X such that

G

⊆

A

. Then

G

c is a

VαOS in X such that c c

G

A

⊆

. Since

A

cis a VGαCS,

V

α

cl

( )

A

c

⊆

G

c. But

V

α

cl

( )

A

c

=

(

V

α

int

( )

A

)

c. Hence

( )

(

V

α

int

A

)

c

⊆

G

. This implies

G

⊆

V

α

int

( )

A

.

Sufficiency: Assume that

G

⊆

V

α

int

( )

A

, whenever G is a VαCS and

G

⊆

A

. Then

(

V

α

int

( )

A

c

)

⊆

G

c

,

whenever

G

cis a VαOS and

V

α

cl

( )

A

c

⊆

G

c

.

Therefore

A

cis a VGαCS. This implies A is a VGαOS.

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